The unramified two-dimensional Langlands correspondence

Abstract

We describe the unramified Langlands correspondence for two-dimensional local fields and construct a categorical analogue of the unramified principal series representation and study its properties. The main tool for this description is the construction of a certain central extension. For this and other central extensions, we prove non-commutative reciprocity laws (that is, splitting of the central extensions over certain subgroups) for arithmetic surfaces and projective surfaces over finite fields. These reciprocity laws connect central extensions constructed locally and globally.